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%   What is a Partial Differential Equation?
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\begin{document}

\mainmatter
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\lectureseries[An Introduction to PDE's]{An Introduction to Partial Differential Equations in the Undergraduate Curriculum}


\auth[Andrew J. Bernoff]{Andrew J. Bernoff} 

%\address{Department of Mathematics, Harvey Mudd College,Claremont, CA 91711}
%\email{ajb@hmc.edu}

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\lecture{What is a Partial Differential Equation?}

\section{Outline of Lecture}
\begin{itemize}
\item{What is a Partial Differential Equation?}
\item{Classifying PDE's: Order, Linear vs. Nonlinear}
\item{Homogeneous PDE's and Superposition}
\item{The Transport Equation} 
\end{itemize}


\section{What is a Partial Differential Equation?}

You've probably all seen an ordinary differential equation (ODE); for example the 
pendulum equation,
\begin{equation}
\frac {d^2 \Theta}{dt^2} + \frac{g}{L} \sin \Theta  =0,
\end{equation}

describes the angle, $\Theta$, a pendulum makes with the vertical as a
function of time, $t$. Here $g$ and $L$ are constants (the
acceleration due to gravity and length of the pendulum respectively),
$t$ is the {\bf independent variable} and $\Theta$ is the 
{\bf dependent variable}. This is an ODE because there is only one
independent variable, here $t$ which represents time.

A partial differential equation (PDE) relates the partial derivatives
of a function of two or more independent variables together.  For
example, Laplace's equation for $\Phi(x,y)$,
\begin{equation}
\frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2}  =0
\label{Laplace}
\end{equation}
arises in many places in mathematics and physics. For simplicity, we
will use subscript notation for partial derivatives, so this equation
can also be written $\Phi_{xx} + \Phi_{yy}=0$.

We say a function is a {\bf solution} to a PDE if it satisfy the
equation and any side conditions given.  Mathematicians are often
interested in if a solution {\bf exists} and when it is {\bf unique}.

\begin{exercise}
Show that $\Phi_1=x$ and $\Phi_2=x^2-y^2$ are solutions to Laplace's
equation (\ref{Laplace}). How can you combine them to create a new
solution?
\end{exercise}

\vfill
\begin{exercise}
Show that 
\begin{equation*}
Z(x,y)= \ln \left ( \frac{\sin(y)}{\sin(x)} \right )
\end{equation*}
is a solution to the {\bf minimal surface equation}, 
\begin{equation}
 (1+Z_y^2)Z_{xx}- 2 Z_x Z_y Z_{xy}+(1+Z_x^2) Z_{yy} =0,
\end{equation}
in the region $0<x< \pi$, $0<y< \pi$. What happens on the boundary of
this region?  Suppose we consider a constant multiple of $Z(x,y)$ --
is it still a solution of the PDE?
\end{exercise}

\vfill
%\begin{center}
%{\it (Room for notes)}
%\end{center}
\eject
\section {Classifying PDE's:  Order, Linear vs. Nonlinear}
When studying ODEs we classify them in an attempt to group similar
equations which might share certain properties, such as methods of
solution.  We classify PDE's in a similar way. The {\bf order} of the
differential equation is the highest partial derivative that appears
in the equation.  So, for example Laplace's Equation (\ref{Laplace})
is second-order.

Some other examples are the {\bf convection equation} for $u(x,t)$,
\begin{equation}
u_t +Cu_x =0,
\label{convection}
\end{equation}
which is first-order. Here $C$ is the wave speed.
The {\bf minimal surface equation}, 
\begin{equation}
 (1+Z_y^2)Z_{xx}- 2 Z_x Z_y Z_{xy}+(1+Z_x^2) Z_{yy} =0,
\label{minimal}
\end{equation}
describes an area minimizing surface, $Z(x,y)$, and is second-order.  Finally, the {\bf Korteweg-deVries equation} (sometimes called {\bf KdV}), 
\begin{equation}
h_t +6hh_x =h_{xxx}
\label{KdV}
\end{equation}
is a model of the amplitude of a wave, $h(x,t)$, on the surface of a fluid and is third-order.

We also define {\bf linear} PDE's as equations for which the dependent
variable (and its derivatives) appear in terms with degree at most
one.  Anything else is called {\bf nonlinear}.  So, for example, the
most general first-order linear PDE for $u(x,t)$ would be
\begin{equation}
a(x,t) u_t + b(x,t) u_x + c(x,t)u =d(x,t) ,
\end{equation}
where $a$, $b$, $c$ and $d$ are known functions (called coefficients).
\begin{exercise}
Which of Laplace's equation (\ref{Laplace}), the convection equation
(\ref{convection}), the minimal surface equation (\ref{minimal}) and
the Korteweg-deVries equation (\ref{KdV}) are linear?
\end{exercise}

\vskip 0.5 in
\begin{exercise}
Write down the most general constant coefficient linear second-order equation for $\Phi(x,y)$.
\end{exercise}

\vfill
\eject

\section {Homogeneous PDE's and Superposition}

Linear equations can further be classified as {\bf homogeneous} for
which the dependent variable (and it derivatives) appear in terms with
degree {\bf exactly} one, and non-homogeneous which may contain terms
which only depend on the independent variable. So, the convection
equation
\begin{equation*}
u_t +cu_x =0
\end{equation*}
is homogeneous, but its cousin, the general first-order linear PDE for $u(x,t)$, is non-homogeneous
\begin{equation*}
a(x,t) u_t + b(x,t) u_x + c(x,t) u =d(x,t) ,
\end{equation*}
unless $d(x,t)=0$.

Because partial differentiation is distributive, you can quickly
convince yourself that if two solutions, say $u_1$ and $u_2$, satisfy
a linear homogeneous PDE, that any linear combination of them
\begin{equation}
u = c_1 u_1 +c_2u_2
\end{equation}
is also a solution. So, for example, since
\begin{equation*}
\Phi_1 = x^2-y^2 \qquad \Phi_2 = x
\end{equation*}
both satisfy Laplace's equation, $\Phi_{xx} + \Phi_{yy}=0$, so does any linear combination of them
\begin{equation*}
\Phi = c_1 \Phi_1 +c_2 \Phi_2 = c_1 (x^2-y^2) + c_2 x .
\end{equation*}
This property is extremely useful for constructing solutions which
satisfy certain initial conditions and boundary conditions.  \vfill
\eject

\section{The Transport Equation} 
One of the driving motivations for studying PDE's is to describe the
physical world around us. We can use a {\bf flux argument} to derive
equations describing the evolution of a {\bf density}, which is just
a fancy word describing the concentration of something (mass in a
region, heat in a metal bar, traffic on a highway) per unit volume.

Consider a one-dimensional freeway and let $\rho(x,t)$ be the density
of cars per unit length on the freeway.

\vskip 3 in

\begin{center}
Figure 1.1: Flux argument  for cars on a freeway.\\
 (draw your own figure).
\end{center}
Then the mass of cars in the region $a<x<b$ is given by
\begin{equation}
M = \int_a^b \rho(x,t) ~ dx ~ .
\end{equation}

Now suppose we are measuring the flux, $Q$, of cars {\bf into} this
region measured in mass/unit time. It can written in terms of the
number of cars crossing into the region at $x=a$, called $q(a)$, minus
the number of cars that flow out of the region at $x=b$, called
$q(b)$,
\begin{equation}
Q = q(a)-q(b) .
\end{equation}
Now, by {\bf conservation of mass}, the rate of change of the mass
between $a$ and $b$ is given by the flux into the region,
\begin{equation}
\frac{dM}{dt} = Q .
\label{CofM}
\end{equation}
We can rewrite the flux by a clever application of the fundamental theorem of calculus:
\begin{equation}
Q = q(a,t)-q(b,t) = - \left. q(x,t) \right  |_{x=a}^{x=b} =- \int_a^b q_x ~dx~.
\end{equation}

We can now rewrite the conservation of mass equation as  
\begin{equation}
\frac{dM}{dt}  = \frac{d}{dt} \int_a^b \rho ~ dx    =  \int_a^b \rho_t  ~ dx    =  Q  =- \int_a^b ~ q_x ~ dx ,
\label{CofM2}
\end{equation}
or, rearranging
\begin{equation}
  \int_a^b \rho_t   +q_x ~ dx =0 .   
\end{equation}
Since this is true for {\bf every} interval $a<x<b$, the integrand must vanish identically. So
\begin{equation}
 \rho_t   +q_x  =0 .   
\end{equation}
Equations of this form are called {\bf transport equations} or 
{\bf conservation laws} -- they are a very active area of study in PDE's.

We can propose a simple model for the flux function $q(x,t)$ --
suppose we assume the cars are all moving at a constant speed $C$.
Then we can argue that the flux is just equal to the product of the
number of cars time the speed they are moving at,
\begin{equation}
q(x,t) = C \rho(x,t) .
\end{equation}

Substituting into the transport equation yields
 \begin{equation}
 \rho_t   + C \rho_x  =0 ,   
 \label{convection2}
\end{equation}
which is just the convection equation.  If we specify the initial distribution of cars,
 \begin{equation}
 \rho(x,0)  = F(x),   
 \label{IC}
\end{equation}
we can show fairly easily that the solution to the convection equation
with this initial condition is just
 \begin{equation}
 \rho(x,t) =F(x-Ct)  ,
\end{equation}
corresponding to cars moving uniformly to the right.
\vfill
\eject
Physically, we just see the distribution of cars translating to the right with a speed of C.
\phantom{top}
\vskip 3 in

\begin{center}
Figure 1.2: Solution to the convection equation.\\
 (draw your own figure).
\end{center}


To verify this solution let $\xi = x-Ct$, and look for a solution $F(\xi)$. Then, by the chain rule
\begin{equation}
F_t = F_\xi \xi_t  = -C F_\xi \qquad F_x = F_\xi \xi_x  = F_\xi
\end{equation}
Substituting $\rho(x,t)=F(\xi)$ into the convection equation (\ref{convection2}), we find
\begin{equation}
 \rho_t   + C \rho_x  =  F_t   + C F_x  = -CF_\xi  + C F_\xi  =0 .   
\end{equation}
Moreover, when $t=0$, we find $\xi=x$ so that the initial condition $\rho(x,0)=F(x)$ is satisfied also.
\vfill
\eject

\section{Challenge Problems for Lecture 1}

\begin{problem}
Classify the follow differential equations as ODE's or PDE's, linear
or nonlinear, and determine their order. For the linear equations,
determine whether or not they are homogeneous.


\begin{enumerate} 
\item The {\bf diffusion equation} for $h(x,t)$: 
$$h_t =D h_{xx}$$ 

\item The {\bf wave equation} for $w(x,t)$:  
$$w_{tt} = c^2 w_{xx}$$

\item The {\bf thin film equation} for $h(x,t)$: 
$$h_t =- (h h_{xxx})_x$$

\item The {\bf forced harmonic oscillator} for $y(t)$:
$$y_{tt} + \omega^2 y = F \cos (\Omega t) $$

\item The {\bf Poisson Equation} for the electric potential $\Phi(x,y,z)$:
$$\Phi_{xx} + \Phi_{yy} +\Phi_{zz}  = 4 \pi \rho(x,y,z) $$
where $\rho(x,y,z)$ is a known charge density.

\item  {\bf Burger's equation} for $h(x,t)$: 
$$h_t + hh_x=\nu  h_{xx}$$ 

\end{enumerate}


\end{problem}

\vfill

\begin{problem}
Suppose when deriving the convection equation, we assumed the speed of the
cars was given by $\beta x$ for $x>0$.
\begin{enumerate}

\item
Explain why the flux function now is given by $q(x,t) = \beta x \rho$ and the 
associated transport equation is given by
\begin{equation*}
\rho_t + \left ( \beta x \rho \right)_x =0 .
\end{equation*}


\item 
Explain why 
\begin{equation*}
\rho(0,t) = 0 , \qquad \rho(x,0) = xe^{-x} 
\end{equation*}
correspond to a {\bf boundary condition} of no flux of cars in from
the origin and an {\bf initial condition} specifying the distribution
of cars at $t=0$.

\item

Verify that 
\begin{equation*}
\rho(x,t)=x e^{-(2 \beta t + x e^{- \beta t})} 
\end{equation*}
is a solution to both the transport equation given in (a) and the initial  and boundary conditions given in (b).

\end{enumerate}


\end{problem}
\vfill
\eject

\begin{problem}
Show that the helicoid
\begin{equation*} 
 Z(x,y)= \tan ^{-1} (y/x) 
\end{equation*}
satisfies the 
minimal surface equation,
\begin{equation*}
 (1+Z_y^2)Z_{xx}- 2 Z_x Z_y Z_{xy}+(1+Z_x^2)Z_{yy}
\end{equation*}
MAPLE may be helpful with the algebra.
\end{problem}

\vfill

\begin{problem}
Show that the soliton
\begin{equation*} 
 h(x,t)= 2 \alpha^2 {\rm sech} \left ( \alpha (x-4\alpha^2 t) \right ) 
\end{equation*}
satisfies the the  Korteweg-deVries equation, 
\begin{equation*}
h_t +6hh_x =h_{xxx}
\end{equation*}
MAPLE may be helpful with the algebra, in particular if you don't remember your 
hyperbolic trigonometric identities.
\end{problem}
\vfill
\eject





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